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Properties

Label 2-570-19.7-c3-0-1
Degree $2$
Conductor $570$
Sign $-0.321 + 0.946i$
Analytic cond. $33.6310$
Root an. cond. $5.79923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s + (−3 + 5.19i)6-s − 19·7-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (−5 + 8.66i)10-s + 15·11-s − 12·12-s + (−25 + 43.3i)13-s + (−19 − 32.9i)14-s + (−7.50 + 12.9i)15-s + (−8 − 13.8i)16-s + (21 + 36.3i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.204 + 0.353i)6-s − 1.02·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + 0.411·11-s − 0.288·12-s + (−0.533 + 0.923i)13-s + (−0.362 − 0.628i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.299 + 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(33.6310\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :3/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5506205153\)
\(L(\frac12)\) \(\approx\) \(0.5506205153\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
19 \( 1 + (66.5 + 49.3i)T \)
good7 \( 1 + 19T + 343T^{2} \)
11 \( 1 - 15T + 1.33e3T^{2} \)
13 \( 1 + (25 - 43.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-21 - 36.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-22.5 + 38.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-54 + 93.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 196T + 2.97e4T^{2} \)
37 \( 1 + 43T + 5.06e4T^{2} \)
41 \( 1 + (106.5 + 184. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (169 + 292. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (120 - 207. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-226.5 + 392. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-90 - 155. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (49 - 84.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (55 - 95.2i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (102 + 176. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-380 - 658. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-107 - 185. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 198T + 5.71e5T^{2} \)
89 \( 1 + (307.5 - 532. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-26 - 45.0i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.80562285169270948462841633812, −9.883072383970417231670499339873, −9.205464819313940605246380569806, −8.380711283718082592470660524679, −7.04028121828366431354255624943, −6.56611998421207075808671927694, −5.49379007846642715055079618938, −4.29926175925290325435219896903, −3.47055270360893680639543755616, −2.26374046104369393761634253107, 0.13285472666336875541499696332, 1.48326773947469385932243162689, 2.80382723753890255913694175730, 3.63582065351854345019735097249, 5.00340450624673127185850464065, 5.98538806538154602295588118269, 6.88972952559080363792953966084, 8.008242755199386979338620774185, 9.047288885128265762125131120285, 9.753018396200894396769828478234

Graph of the $Z$-function along the critical line